Physical Chemistry Third Edition

17.3 The Radial Factor in the Hydrogen Atom Wave Function. The Energy Levels of the Hydrogen Atom 737

where we now omit the subscript “rel” fromErel. The radial factorR(r) is different for

each choice of the potential energy functionV(r). In a later chapter we will choose

a representation forV(r) that will allow us to use Eq. (17.3-2) for the rotation and

vibration of a diatomic molecule. To apply it to the hydrogen atom, we substitute the

expression for the potential energy given in Eq. (17.1-1) into this equation, and expand

the derivative term into two terms:

−r^2

d^2 R

dr^2

− 2 r

dR

dr

−

2 μr^2

h ̄^2

(

E+

e^2

4 πε 0 r

)

R+l(l+1)R 0 (17.3-3)

We make the following substitutions:

α^2 −

2 μE

h ̄^2

, β

μe^2

4 πε 0 αh ̄^2

, ρ 2 αr (17.3-4)

When the resulting equation is divided byρ^2 we obtain an equation that is known as

theassociated Laguerre equation:

d^2 R

dρ^2

+

2

ρ

dR

dρ

−

R

4

+

βR

ρ

−l(l+1)

R

ρ^2

 0 (17.3-5)

whereR(ρ) is the function ofρthat is equal toR(r).

Exercise 17.5

Carry out the manipulations to obtain Eq. (17.3-5) from Eq. (17.3-3).

This equation is named for Edmund
Laguerre, 1834–1866, the famous
French mathematician who solved it.

Laguerre assumed that the solution of the associated Laguerre equation can be

written as

R(ρ)G(ρ)e−ρ/^2 (17.3-6)

whereG(ρ) is a power series

G(ρ)

∑∞

j 0

ajρj (17.3-7)

with constant coefficientsa 1 ,a 2 ,a 3 ,.... We do not discuss the solution that leads to

expressions for these coefficients. As with the harmonic oscillator equation, the series

must terminate to satisfy the boundary condition that the wave function is finite.^1 This

turns the seriesGinto a polynomial.

The termination of the seriesGturns out to require that the parameterβis equal to

an integer that must be at least as large asl+1.^2 We denote this integer byn. Solving

the second equality in Eq. (17.3-4) forα, we obtain

α

μe^2

4 πε 0 h ̄^2 β



μe^2

4 πε 0 h ̄^2 n

(17.3-8)

(^1) F. L. Pilar,Elementary Quantum Chemistry, McGraw-Hill, New York, 1968, p. 151ff.
(^2) I. N. Levine,Quantum Chemistry, 6th ed., Prentice-Hall, Upper Saddle River, N.J., 2000, p. 136ff.